Chapter 8 Flow in Conduits

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1 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall Chapter 8 Flow in Conduits Entrance and developed flows Le = f(d, V,, ) i theorem Le/D = f(re) Laminar flow: Re crit 000, i.e., for Re < Re crit laminar Re > Re crit turbulent Le/D =.06Re from experiments Le max =.06Re crit D 138D maximum Le for laminar flow

2 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall 013 Turbulent flow: Le 1/ 4.4Re 6 D from experiment Re Le/D i.e., relatively shorter than for laminar flow Laminar vs. Turbulent Flow laminar turbulent spark photo Reynolds 1883 showed difference depends on Re = VD

3 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall Shear-Stress Distribution Across a Pipe Section Continuity: Q 1 = Q = constant, i.e., V 1 = V since A 1 = A Momentum: F s uv A = V1 V1 A1 V V A = V V 0 pa p dp ds W Ads Ads r Q 1 dsa Wsin dz sin ds d ds dp ds dsa Ads p z rds 0 dz ds rds 0 r0 d p z W ds varies linearly from 0.0 at r = 0 (centerline) to max (= w ) at r = r 0 (wall), which is valid for laminar and turbulent flow.

4 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall Laminar Flow in Pipes dv dy dv dr r d ds p z y = wall coordinate = r o r dv dr r d ds p z dv dr dv dy dy dr dv dy V r 4 d ds p z C o 0 C p z V r o no slip condition r 4 d ds r o r d V r p z VC 1 r 4 ds r 0 where Q VdA r o = V r 0 ro d VC p z 4 ds rdr Exact solution to Navier-Stokes equations for laminar flow in circular pipe da rdrd rdr

5 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall Q 4 r 8 d ds o o p z V p z Q r d VC A 8 ds For a horizontal pipe, ro d ro p VC p z 4 ds 4 L where L = length of pipe = ds r p r p V r r r 4 L r0 4L 4 r0 p D p Q r0 r rdr 0 L 18L o 1 0 Energy equation: p 1 V1 p V z1 z h L g g p1 p h z1 z

6 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall p p dh L d L W ( ) [ ] [ ] ds ds r 1 hl z1 z h L p z 8w Define friction factor f V friction coefficient for pipe flow 8 L W L f V L V h L h f [ ] [ ] f r r D g 0 0 C f 0 w 1 V boundary layer flow Darcy Weisbach Equation, which is valid for both laminar and turbulent flow. Friction factor definition based on turbulent flow analysis 8 where w w( r o, V,,, k ) thus n=6, m=3 and r=3 such that w i=1,,3 = f, V Re VD VD, k/d; or f=f(re, k/d) where k=roughness height. For turbulent flow f determined from turbulence modeling since exact solutions not known, as will be discussed next. For laminar flow f not affected k and f(re) determined from exact analytic solution to Navier-Stokes equations. Exact solution: w ro d r o 8V 4V p z = = ds ro ro

7 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall For laminar flow w w( r o, V, ) thus n=4, m=3 and r=1 such that W o =constant. The constant depends on duct shape 1 r V (circular, rectangular, etc.) and is referred to as Poiseuille number=p o. P o =4 for circular duct. or f ro V VD Re 3LV hf h L h f = head loss due to friction D for z=0: p V as per Hagen!

8 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall Stability and Transition Stability: can a physical state withstand a disturbance and still return to its original state. In fluid mechanics, there are two problems of particular interest: change in flow conditions resulting in (1) transition from one to another laminar flow; and () transition from laminar to turbulent flow. (1) Example of transition from one to another laminar flow: Centrifugal instability for Couette flow between two rotating cylinders when centrifugal rc force > viscous force 3 i i o Ta c r r r, Ta cr =1708 ( 0 i i) which is predicted by small-disturbance/linear stability theory.

9 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall () Transition from laminar to turbulent flow Not all laminar flows have different equilibrium states, but all laminar flows for sufficiently large Re become unstable and undergo transition to turbulence. Transition: change over space and time and Re range of laminar flow into a turbulent flow. U Re ~ 1000, δ = transverse viscous thickness cr Re trans > Re cr with x trans ~ 10-0 x cr Small-disturbance/linear stability theory also predicts Re cr with some success for parallel viscous flow such as plane Couette flow, plane or pipe Poiseuille flow, boundary layers without or with pressure gradient, and free shear flows (jets, wakes, and mixing layers). No theory for transition, but recent Direct Numerical Simualtions is helpful. In general: Re trans =Re trans (geometry, Re, pressure gradient/velocity profile shape, free stream turbulence, roughness, etc.)

10 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall Criterion for Laminar or Turbulent Flow in a Pipe Re crit 000 Re trans 3000 Re = V D/ flow becomes unstable flow becomes turbulent Turbulent Flow in Pipes Continuity and momentum: r d o W ds o r r p z L d ds Energy: h f Combining: h f h f L W r o p z 1 V 8 W define f = friction factor L g r o 1 V 8 f h f L V f Darcy Weisbach Equation D g f = f(re, k/d) = still must be determined! Re VD k = roughness

11 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall Description of Turbulent Flow Most flows in engineering are turbulent: flows over vehicles (airplane, ship, train, car), internal flows (heating and ventilation, turbo-machinery), and geophysical flows (atmosphere, ocean). V (x, t) and p(x, t) are random functions of space and time, but statistically stationary flaws such as steady and forced or dominant frequency unsteady flows display coherent features and are amendable to statistical analysis, i.e. time and space (conditional) averaging. RMS and other low-order statistical quantities can be modeled and used in conjunction with averaged equations for solving practical engineering problems. Turbulent motions range in size from the width in the flow δ to much smaller scales, which come progressively smaller as the Re = Uδ/υ increases.

12 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall 013 1

13 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall Physical description: (1) Randomness and fluctuations: Turbulence is irregular, chaotic, and unpredictable. However, for statistically stationary flows, such as steady flows, can be analyzed using Reynolds decomposition. u t0 T 1 u u' u u dt T t0 t0 T u ' 0 u 1 u' dt T t0 ' etc. u = mean motion u ' = superimposed random fluctuation u ' = Reynolds stresses; RMS = u ' Triple decomposition is used for forced or dominant frequency flows u u u' ' u' Where u '' = organized component () Nonlinearity Reynolds stresses and 3D vortex stretching are direct result of nonlinear nature of turbulence. In fact, Reynolds stresses arise from nonlinear convection term after substitution of Reynolds decomposition into NS equations and time averaging.

14 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall (3) Diffusion Large scale mixing of fluid particles greatly enhances diffusion of momentum (and heat), i.e., Reynolds Stresses: u' u' i j Isotropic eddy viscosity: u ' u ' viscous stress ij k 3 i j t ij ij (4) Vorticity/eddies/energy cascade Turbulence is characterized by flow visualization as eddies, which vary in size from the largest L δ (width of flow) to the smallest. The largest eddies have velocity scale U and time scale L δ /U. The orders of magnitude of the smallest eddies (Kolmogorov scale or inner scale) are: L K = Kolmogorov micro-scale = 1 3 L 4 ( v / ) 3 U L K = O(mm) >> L mean free path = 6 x 10-8 m Velocity scale = (νε) 1/4 = O(10 - m/s) Time scale = (ν/ε) 1/ = O(10 - s) 3 1/ 4 Largest eddies contain most of energy, which break up into successively smaller eddies with energy transfer to yet smaller eddies until L K is reached and energy is dissipated at rate by molecular viscosity. ij

15 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall Richardson (19): L δ Big whorls have little whorls Which feed on their velocity; And little whorls have lesser whorls, L K And so on to viscosity (in the molecular sense). (5) Dissipation L u 0 0 Re k 0( U ) u 0 0 k u' / big v' w' Energy comes from largest scales and fed by mean motion ε = rate of dissipation = energy/time u0 o o 0 u0 3 u = 0 3 independent υ L l K 0 The mathematical complexity of turbulence entirely precludes any exact analysis. A statistical theory is well developed; however, it is both beyond the scope of this course and not generally useful as a predictive tool. Since the time of Reynolds (1883) turbulent flows have been analyzed by considering the mean (time averaged) motion 1 4 Dissipation occurs at smallest scales

16 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall and the influence of turbulence on it; that is, we separate the velocity and pressure fields into mean and fluctuating components. It is generally assumed (following Reynolds) that the motion can be separated into a mean (u, v, w, p) and superimposed turbulent fluctuating (u, v, w, p ) components, where the mean values of the latter are 0. u u u p p p v v v and for compressible flow w w w and T T T where (for example) t 0 t 1 1 u udt t 1 t 0 and t 1 sufficiently large that the average is independent of time Thus by definition u 0, etc. Also, note the following rules which apply to two dependent variables f and g f f f g f g f g f g f s f s fds fds f = (u, v, w, p) s = (x, y, z, t) The most important influence of turbulence on the mean motion is an increase in the fluid stress due to what are

17 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall called the apparent stresses. Also known as Reynolds stresses: ij uu i j u uv uw = uv uw vw v vw w Symmetric nd order tensor The mean-flow equations for turbulent flow are derived by substituting V V V into the Navier-Stokes equations and averaging. The resulting equations, which are called the Reynolds-averaged Navier-Stokes (RANS) equations are: Continuity V 0 i.e. V 0 and V 0 DV i j gkˆ Dt x j Momentum u u p V or DV Dt ij gkˆ p u x i j u x j i ij ui uj ij u 1 = u u = v u 3 = w x 1 = x x = y x 3 = z

18 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall Comments: 1) equations are for the mean flow ) differ from laminar equations by Reynolds stress terms = uu i j 3) influence of turbulence is to transport momentum from one point to another in a similar manner as viscosity 4) since u iuj are unknown, the problem is indeterminate: the central problem of turbulent flow analysis is closure! 4 equations and = 10 unknowns

19 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall

20 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall Turbulence Modeling Closure of the turbulent RANS equations require the determination of uv, etc. Historically, two approaches were developed: (a) eddy viscosity theories in which the Reynolds stresses are modeled directly as a function of local geometry and flow conditions; and (b) mean-flow velocity profile correlations, which model the mean-flow profile itself. The modern approaches, which are beyond the scope of this class, involve the solution for transport PDE s for the Reynolds stresses, which are solved in conjunction with the momentum equations. (a) eddy-viscosity: theories u uv t y The problem is reduced to modeling t, i.e., t = t (x, flow at hand) Various levels of sophistication presently exist in modeling t V L turbulent velocity scale The total stress is t total molecular viscosity t t t u y In analogy with the laminar viscous stress, i.e., t mean-flow rate of strain turbulent length scale eddy viscosity (for high Re flow t >> ) where V t and L t are based on large scale turbulent motion

21 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall Mixing-length theory (Prandtl, 190) uv c u v based on kinetic theory of gases u 1 v u y u y 1 and are mixing lengths which are analogous to molecular mean free path, but much larger Known as a zero equation model since no additional PDE s are solved, only an algebraic relation uv u y t u y y = f(boundary layer, jet, wake, etc.) Although mixing-length theory has provided a very useful tool for engineering analysis, it lacks generality. Therefore, more general methods have been developed. One and two equation models distance across shear layer t Ck

22 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall 013 C = constant k = turbulent kinetic energy = u v w = turbulent dissipation rate Governing PDE s are derived for k and which contain terms that require additional modeling. Although more general than the zero-equation models, the k- model also has definite limitation; therefore, relatively recent work involves the solution of PDE s for the Reynolds stresses themselves. Difficulty is that these contain triple correlations that are very difficult to model. Most recent work involves direct and large eddy simulation of turbulence. (b) mean-flow velocity profile correlations As an alternative to modeling the Reynolds stresses one can model mean flow profile directly for wall bounded flows such as pipes/channels and boundary layers. For simple - D flows this approach is quite good and will be used in this course. For complex and 3-D flows generally not successful. Consider the shape of a turbulent velocity profile for wall bounded flow.

23 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall Note that very near the wall laminar must dominate since u i u j = 0 at the wall (y = 0) and in the outer part turbulent stress will dominate. This leads to the three-layer concept: Inner layer: Outer layer: viscous stress dominates turbulent stress dominates

24 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall Overlap layer: both types of stress important 1. laminar sub-layer (viscous shear dominates) From dimensional analysis u = f(, w,, y) note: not f() and =D and y = r o r for pipe flow u f y law-of-the-wall u u u where: * u * = friction velocity = w / y * yu very near the wall: w constant = i.e., du u cy dy y u 0 < y + < 5

25 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall outer layer (turbulent shear dominates) ue u g, w,, y outer dp note: independent of and actually also depends on dx From dimensional analysis ue u y g * u velocity defect law 3. overlap layer (viscous and turbulent shear important) It is not that difficult to show that for both laws to overlap, f and g are logarithmic functions: Inner region: du u df dy dy Outer region: du u dg dy d u y u df dy u y u dg ; valid at large y d + and small η. f(y+) g(η)

26 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall Therefore, both sides must equal universal constant, 1 1 f ( y ) ln y B u / u 1 ue u g( ) ln A u (inner variables) (outer variables), A, and B are pure dimensionless constants Values vary somewhat depending on different exp. arrangements = 0.41 Von Karman constant B = 5.5 (or 5.0) A =.35 BL flow = 0.65 pipe flow The difference is due to loss of intermittency in duct flow. A = 0 means small outer layer

27 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall 013 7

28 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall 013 8

29 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall Note that the y + scale is logarithmic and thus the inner law only extends over a very small portion of Inner law region <. And the log law encompasses most of the pipe/boundarylayer. Thus as an approximation one can simply assume u u * 1 ln y B u * / w yu * y is valid all across the shear layer. This is the approach used in this course for turbulent flow analysis. The approach is a good approximation for simple and -D flows (pipe and flat plate), but does not work for complex and 3-D flows.

30 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall

31 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall Velocity Distribution and Resistance in Smooth Pipes Assume log-law is valid across entire pipe * u r 1 r ln o r u B * u u * w.41 B 5.0 friction velocity r o rdr * u r Q 1 ru 3 0 * o V u ln B A ro drop over bar: u * 1/ rou V.44ln 1.34 * o f 1/ V 1 Re f 8 8 1/ f 1.99log 1/ Re f constants adjusted using data 1 1/ log Re f. 8 f Re > 3000 Since f equation is implicit, it is not easy to see dependency on ρ, μ, V, and D f ( pipe) Re 1/ 4 D 4000 < Re D < 10 5 Blasius (1911) power law curve fit to data

32 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall p L V hf f D g p 0.158L D V 3/ 4 1/ 4 5/ 4 7/ 4 Near quadratic (as expected) Nearly linear Only slightly with μ Drops weakly with pipe size 3/ 4 1/ L D Q 1.75 laminar flow: p 8LQ / R 4 p (turbulent) increases more sharply than p (laminar) for same Q; therefore, increase D for smaller p. D decreases p by 7 for same Q. u * max ur ( 0) 1 0 * * ln ru u u B Combine with V u Ru * 1 * ln B 3 V u 1 1. f 1 max

33 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall Power law fit to velocity profile: u u max r 1 ro m = m(re) m

34 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall Viscous Distribution and Resistance Rough Pipes For laminar flow, effect of roughness is small; however, for turbulent flow the effect is large. Both laminar sublayer and overlap layer are affected. Inner layer: u = u (y, k,, w ) u + = u + (y/k) not function of as was case for smooth pipe (or wall) Outer layer: unaffected Overlap layer: 1 y u R ln constant rough k 1 u S ln y B smooth * 1 ku us u R ln k constant k B(k + ) i.e., rough-wall velocity profile shifts downward by B(k + ), which increases with k +. Three regions of flow depending on k + 1. k + < 5 hydraulically smooth (no effect of roughness). 5 < k + < 70 transitional roughness (Re dependence) 3. k + > 70 fully rough (independent Re)

35 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall For 3, B ln k 3. 5 from data u u V 1 ln y k D.44ln k 8.5 f * 3. Re fully rough flow f 1 1/ log k / D 3.7 Composite Log-Law Smooth wall log law u B * 1 ln y B B k 1 5 ln 1.3k B * from data f 1 1/ k / D.51 log 1/ 3.7 Ref Moody Diagram = 1.14 k s 9.35 log 1/ D Ref

36 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall

37 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall

38 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall

39 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall There are basically three types of problems involved with uniform flow in a single pipe: 1. Determine the head loss, given the kind and size of pipe along with the flow rate, Q = A*V. Determine the flow rate, given the head, kind, and size of pipe 3. Determine the pipe diameter, given the type of pipe, head, and flow rate 1. Determine the head loss The first problem of head loss is solved readily by obtaining f from the Moody diagram, using values of Re and k s /D computed from the given data. The head loss h f is then computed from the Darcy-Weisbach equation. f = f(re D, k s /D) LV hf f h D g p = z p1 p h z1 z Re D = Re D (V, D). Determine the flow rate The second problem of flow rate is solved by trial, using a successive approximation procedure. This is because both Re and f(re) depend on the unknown velocity, V. The solution is as follows:

40 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall ) solve for V using an assumed value for f and the Darcy-Weisbach equation gh f V L/ D 1/ f 1/ known from given data note sign ) using V compute Re 3) obtain a new value for f = f(re, k s /D) and reapeat as above until convergence 3/ 1/ D Or can use Re f scale on Moody Diagram gh L f 1/ 1/ f 1) compute Re and k s /D ) read f L V 3) solve V from h f f D g 4) Q = VA 3. Determine the size of the pipe The third problem of pipe size is solved by trial, using a successive approximation procedure. This is because h f, f, and Q all depend on the unknown diameter D. The solution procedure is as follows:

41 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall ) solve for D using an assumed value for f and the Darcy-Weisbach equation along with the definition of Q 8LQ D gh f 1/5 known from given data f 1/5 ) using D compute Re and k s /D 3) obtain a new value of f = f(re, k s /D) and reapeat as above until convergence Flows at Pipe Inlets and Losses From Fittings For real pipe systems in addition to friction head loss these are additional so called minor losses due to 1. entrance and exit effects. expansions and contractions 3. bends, elbows, tees, and other fittings 4. valves (open or partially closed) can be large effect For such complex geometries we must rely on experimental data to obtain a loss coefficient

42 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall In general, h m head loss due to minor losses V g K K = K(geometry, Re, /D) dependence usually not known Loss coefficient data is supplied by manufacturers and also listed in handbooks. The data are for turbulent flow conditions but seldom given in terms of Re. Modified Energy Equation to Include Minor Losses (where V V ): p 1 1 p 1 z1 1V1 h p z V h t hf g g h h m V K g Note: h m does not include pipe friction and e.g. in elbows and tees, this must be added to h f. m

43 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall Flow in a bend: 1 v r 1 p r centrifugal acceleration p i.e. 0 which is an adverse pressure gradient in r r direction. The slower moving fluid near wall responds first and a swirling flow pattern results. This swirling flow represents an energy loss which must be added to the h L. Also, flow separation can result due to adverse longitudinal pressure gradients which will result in additional losses.

44 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall This shows potential flow is not a good approximate in internal flows (except possibly near entrance). Valves: enormous losses 3. Entrances: depends on rounding of entrance 4. Exit (to a large reservoir): K = 1 i.e., all velocity head is lost 5. Contractions and Expansions sudden or gradual theory for expansion: h L V 1 V g d D from continuity, momentum, and energy

45 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall (assuming p = p 1 in separation pockets) h m KSE V1 d 1 D g no theory for contraction: K SC d.4 1 D from experiment If the contraction or expansion is gradual the losses are quite different. A gradual expansion is called a diffuser. Diffusers are designed with the intent of raising the static pressure. C p p p 1 V 1 1 Cp ideal A 1 A 1 Bernoulli and continuity equation K h m C p C ideal p Energy equation V g

46 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall Actually very complex flow and C p = C p (geometry, inlet flow conditions) i.e., fully developed (long pipe) reduces C p thin boundary layer (short pipe) high C p (more uniform inlet profile) Turbulent flow K =.5

47 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall See textbook Table 8. for a table of the loss coefficients for pipe components

48 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall

49 57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall

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